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Please reason step by step, and put your final answer within oxed{}. | [
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"value": ". The operation $\\otimes$ is defined for all nonzero numbers by $a\\otimes b =\\frac{a^{2}}{b}$. Determine $[(1\\otimes 2)\\otimes 3]-[1\\otimes (2\\otimes 3)]$.\n$\\text{(A)}\\ -\\frac{2}{3}\\qquad\\text{(B)}\\ -\\frac{1}{4}\\qquad\\text{(C)}\\ 0\\qquad\\text{(D)}\\ \\frac{1}{4... |
Please reason step by step, and put your final answer within oxed{}. | [
{
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"value": ". Doug constructs a square window using $8$ equal-size panes of glass, as shown. The ratio of the height to width for each pane is $5 : 2$, and the borders around and between the panes are $2$ inches wide. In inches, what is the side length of the square window?\n[asy] fill((0,0)... |
Please reason step by step, and put your final answer within oxed{}. | [
{
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"value": ". $P(x)$ is a polynomial of degree $3n$ such that\n\\begin{eqnarray*} P(0) = P(3) = \\cdots &=& P(3n) = 2, \\\\ P(1) = P(4) = \\cdots &=& P(3n-2) = 1, \\\\ P(2) = P(5) = \\cdots &=& P(3n-1) = 0, \\quad\\text{ and }\\\\ && P(3n+1) = 730.\\end{eqnarray*}\nDetermine $n$."
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Please reason step by step, and put your final answer within oxed{}. | [
{
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"value": ". Let $f$ be the function defined by $f(x)=ax^2-\\sqrt{2}$ for some positive $a$. If $f(f(\\sqrt{2}))=-\\sqrt{2}$ then $a=$\n$\\text{(A) } \\frac{2-\\sqrt{2}}{2}\\quad \\text{(B) } \\frac{1}{2}\\quad \\text{(C) } 2-\\sqrt{2}\\quad \\text{(D) } \\frac{\\sqrt{2}}{2}\\quad \\text{(E... |
Please reason step by step, and put your final answer within oxed{}. | [
{
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"value": ". The total in-store price for an appliance is $\\textdollar 99.99$. A television commercial advertises the same product for three easy payments of $\\textdollar 29.98$ and a one-time shipping and handling charge of $\\textdollar 9.98$. How many cents are saved by buying the appl... |
Please reason step by step, and put your final answer within oxed{}. | [
{
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"value": ". Points $A,B,C,D,E$ and $F$ lie, in that order, on $\\overline{AF}$, dividing it into five segments, each of length 1. Point $G$ is not on line $AF$. Point $H$ lies on $\\overline{GD}$, and point $J$ lies on $\\overline{GF}$. The line segments $\\overline{HC}, \\overline{JE},$ a... |
Please reason step by step, and put your final answer within oxed{}. | [
{
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"value": ". During the softball season, Judy had $35$ hits. Among her hits were $1$ home run, $1$ triple and $5$ doubles. The rest of her hits were single. What percent of her hits were single?\n$\\text{(A)}\\ 28\\% \\qquad \\text{(B)}\\ 35\\% \\qquad \\text{(C)}\\ 70\\% \\qquad \\text{... |
Please reason step by step, and put your final answer within oxed{}. | [
{
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"value": ". The graph, $G$ of $y=\\log_{10}x$ is rotated $90^{\\circ}$ counter-clockwise about the origin to obtain a new graph $G'$. Which of the following is an equation for $G'$?\n(A) $y=\\log_{10}\\left(\\frac{x+90}{9}\\right)$ (B) $y=\\log_{x}10$ (C) $y=\\frac{1}{x+1}$ (D) $y=10^{-x}$... |
Please reason step by step, and put your final answer within oxed{}. | [
{
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"value": ". Jose, Thuy, and Kareem each start with the number 10. Jose subtracts 1 from the number 10, doubles his answer, and then adds 2. Thuy doubles the number 10, subtracts 1 from her answer, and then adds 2. Kareem subtracts 1 from the number 10, adds 2 to his number, and then dou... |
Please reason step by step, and put your final answer within oxed{}. | [
{
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"value": ". A set of consecutive positive integers beginning with $1$ is written on a blackboard. One number is erased. The average (arithmetic mean) of the remaining numbers is $35\\frac{7}{17}$. What number was erased? \n$\\textbf{(A)}\\ 6\\qquad \\textbf{(B)}\\ 7 \\qquad \\textbf{(C)}... |
Please reason step by step, and put your final answer within oxed{}. | [
{
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"value": ". A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape, as shown. The parallel sides of the trapezoid have lengths $15$ and $25$ meters. What fraction of the yard is occupied by the flow... |
Please reason step by step, and put your final answer within oxed{}. | [
{
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"value": ". What is the sum of all possible values of $t$ between $0$ and $360$ such that the triangle in the coordinate plane whose vertices are \\[(\\cos 40^\\circ,\\sin 40^\\circ), (\\cos 60^\\circ,\\sin 60^\\circ), \\text{ and } (\\cos t^\\circ,\\sin t^\\circ)\\]\nis isosceles? \n$\\te... |
Please reason step by step, and put your final answer within oxed{}. | [
{
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"value": ". Given: $x > 0, y > 0, x > y$ and $z\\ne 0$. The inequality which is not always correct is:\n$\\textbf{(A)}\\ x + z > y + z \\qquad\\textbf{(B)}\\ x - z > y - z \\qquad\\textbf{(C)}\\ xz > yz$\n$\\textbf{(D)}\\ \\frac {x}{z^2} > \\frac {y}{z^2} \\qquad\\textbf{(E)}\\ xz^2 > yz^2... |
Please reason step by step, and put your final answer within oxed{}. | [
{
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"value": ". The harmonic mean of two numbers can be calculated as twice their product divided by their sum. The harmonic mean of $1$ and $2016$ is closest to which integer?\n$\\textbf{(A)}\\ 2 \\qquad \\textbf{(B)}\\ 45 \\qquad \\textbf{(C)}\\ 504 \\qquad \\textbf{(D)}\\ 1008 \\qquad \\tex... |
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